Past Functional Analysis Seminar

This talk will introduce some of the basic notions and results in the theory of C_0-semigroups, including generation theorems, growth and spectral bounds. If time permits, I will also try to discuss one or two classical results in the asymptotic theory of C_0-semigroups.

Pick's theorem is a century-old theorem in complex analysis about interpolation with bounded analytic functions. The Kadison-Singer problem was a question about states on $C^*$-algebras originating in the work of Dirac on the mathematical description of quantum mechanics. It was solved by Marcus, Spielman and Srivastava a few years ago.

I will talk about Pick's theorem, the Kadison-Singer problem and how the two can be brought together to solve interpolation problems with infinitely many nodes. This talk is based on joint work with Alexandru Aleman, John McCarthy and Stefan Richter.

There is a classical mathematical theorem (due to Banach and Tarski) that implies the following shocking statement: An orange can be divided into finitely many pieces, these pieces can be rotated and rearranged in such a way to yield two oranges of the same size as the original one. In 1929 J.~von Neumann recognizes that one of the reasons underlying the Banach-Tarski paradox is the fact that on the unit ball there is an action of a discrete subgroup of isometries that fails to have the property of amenability ("Mittelbarkeit" in German).

In this talk we will address more recent developments in relation to the dichotomy amenability vs. existence of paradoxical decompositions in different mathematical situations like, e.g., for metric spaces, for algebras and operator algebras. We will present a result unifying all these approaches in terms of a class of C*-algebras, the so-called uniform Roe algebras.

P. Ara, K. Li, F. Lledó and J. Wu, Amenability of coarse spaces and K-algebras , Bulletin of Mathematical Sciences 8 (2018) 257-306;
P. Ara, K. Li, F. Lledó and J. Wu, Amenability and uniform Roe algebras, Journal of Mathematical Analysis and Applications 459 (2018) 686-716;

There is a class $\mathcal{B}$ of analytic Besov functions on a half-plane, with a very simple description.   This talk will describe a bounded functional calculus $f \in \mathcal{B} \mapsto f(A)$ where $-A$ is the generator of either a bounded $C_0$-semigroup on Hilbert space or a bounded analytic semigroup on a Banach space.    This calculus captures many known results for such operators in a unified way, and sometimes improves them.   A discrete version of the functional calculus was shown by Peller in 1983.

Let $X$ be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their $X$-by-$X$ matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of $X$. This algebra "encodes" the large-scale (a.k.a. coarse) structure of $X$. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and an expander-ish condition on graphs that came out of trying to find a counterexample. (Joint with: A. Tikuisis, J. Zhang, K. Li and P. Nowak.)
 

Let $\mathcal{B}$ be a (unital) commutative Banach algebra and $\Omega$ the set of non-trivial multiplicative linear functionals $\omega : \mathcal{B} \to \mathbb{C}$. Gelfand theory tells us that the kernels of these functionals are exactly the maximal ideals of $\mathcal{B}$ and, as a consequence, an element $b \in \mathcal{B}$ is invertible if and only if $\omega(b) \neq 0$ for all $\omega \in \Omega$. A generalisation to non-commutative Banach algebras is the local principle of Allan and Douglas, also known as central localisation: Let $\mathcal{B}$ be a Banach algebra, $Z$ a closed subalgebra of the center of $\mathcal{B}$ and $\Omega$ the set of maximal ideals of $Z$. For every $\omega \in \Omega$ let $\mathcal{I}_{\omega}$ be the smallest ideal of $\mathcal{B}$ which contains $\omega$. Then $b \in \mathcal{B}$ is invertible if and only if $b + \mathcal{I}_{\omega}$ is invertible in $\mathcal{B} / \mathcal{I}_{\omega}$ for every $\omega \in \Omega$.

From an operator theory point of view, one of the most important features of the local principle is the application to Calkin algebras. In that case the invertible elements are called Fredholm operators and the corresponding spectrum is called the essential spectrum. Therefore, by taking suitable subalgebras, we can obtain a characterisation of Fredholm operators. Many beautiful results in spectral theory, e.g.~formulas for the essential spectrum of Toeplitz operators, can be obtained in this way. However, the central localisation is often not sufficient to provide a satisfactory characterisation for more general operators. In this talk we therefore consider a generalisation where the ideals $\mathcal{I}_{\omega}$ do not originate from the center of the algebra. More precisely, we will start with general $L^p$-spaces and apply limit operator methods to obtain a Fredholm theory that is applicable to many different settings. In particular, we will obtain characterisations of Fredholmness and compactness in many new cases and also rediscover some classical results.

This talk is based on joint work with Christian Seifert.

 Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve  pointwise conditions for the functions and their derivatives.  This implies that to prove   regularity results for an operator among these spaces  we need its pointwise expression.  In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$  in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)

In  the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of  the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.

 In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.

We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.

C*-algebras constructed from topological groupoids allow us to study many interesting and a priori very different constructions
of C*-algebras in a common framework. Moreover, they are general enough to appear intrinsically in the theory. In particular, it was recently shown
by Xin Li that all C*-algebras falling within the scope of the classification program admit (twisted) groupoid models.
In this talk I will give a gentle introduction to this class of C*-algebras and discuss some of their structural properties, which appear in connection
with the classification program.
 

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities.  Having its roots in a  pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.

The Schur product is the commutative operation of entrywise

multiplication of two (possibly infinite) matrices. If we fix a matrix

A and require that the Schur product of A with the matrix of any

bounded operator is again the matrix of a bounded operator, then A is

said to be a Schur multiplier; Schur multiplication by A then turns

out to be a completely bounded map. The Schur multipliers were

characterised by Grothendieck in the 1950s. In a 2006 paper, Kissin

and Shulman study a noncommutative generalisation which they call

"operator multipliers", in which the theory of operator spaces plays

an important role. We will present joint work with Katja Juschenko,

Ivan Todorov and Ludmilla Turowska in which we determine the operator

multipliers which are completely compact (that is, they satisfy a

strengthening of the usual notion of compactness which is appropriate

for completely bounded maps).